Quantum field theory for classical fields

TL;DR

This paper develops a framework that maps classical field theories with probabilistic initial conditions to quantum field theories using statistical observables, highlighting the quantum nature of fluctuations.

Contribution

It introduces a novel approach to describe classical fields with fluctuations as quantum field theories through statistical observables and constructs the corresponding functional integral.

Findings

Classical probabilistic fields can be reformulated as quantum field theories.

Operators associated with observables are non-commuting, following quantum rules.

The approach is applied to the relativistic Klein-Gordon equation with interactions.

Abstract

For classical field theories with probabilistic initial conditions the classical field observables are an idealization. Their arbitrarily precise values poorly reflect the characteristic uncertainty in the presence of substantial fluctuations. We propose to describe this system by observables based on fluctuating fields. In terms of these "statistical observables" the probabilistic classical field theory becomes a quantum field theory. Non-commuting operators are associated to observables. The quantum rules follow from the laws for classical probabilities. We construct the functional integral for the quantum field theory, and discuss in detail the classical relativistic Klein-Gordon equation with interactions.